Cos 45 Degrees: Value of cos 45 with proof, Examples and FAQ

Cos 45 degrees

In a 45 degrees right-angled triangle, the cosine of angle 45° is a value representing the ratio of the length of the adjacent side to the length of the hypotenuse (of angle θ).

In trigonometry, we write the cosine of angle 45° mathematically, and its exact value in fraction form is 1/√2. Therefore, we write it in the following form in trigonometry.

cos (45°) = cos π/4 = 1/√2

Value of Cos 45°

The exact value of the cosine of angle 45 degrees is 1/√2 equal to 0.7071067812… in decimal form. The approximate value of the cosine of angle 45 is equal to 0.7071.

Cos (45°) = 0.7071067812… ≈ 0.7071

Proof

The exact value of cos 45 can be derived using three methods. We will use them one by one.

• Theoretical Method

According to the right-angled triangle property, the lengths of the sides adjacent and opposite to the angle θ are equal when the angle of the right-angled triangle is equal to 45°. 

Take, the length of both adjacent side (BC) and opposite side (AB) as ‘l’ (as the right-angled triangle is isosceles when one of its angles is 45°) and the length of hypotenuse as ‘r’. Now, according to the Pythagoras theorem, we know that 

\text { Hypotenuse }^{2}=\text { Perpendicular }^{2}+\text { Adjacent Side }^{2}

\text { So, } A C^{2}=A B^{2}+B C^{2}

\Rightarrow \mathrm{r}^{2}=\mathrm{l}^{2}+\mathrm{l}^{2}
\Rightarrow \mathrm{r}^{2}=2 \mathrm{l}^{2}
\Rightarrow \mathrm{r}=\sqrt{2} \mathrm{l}
\Rightarrow \mathrm{l} / \mathrm{r}=1 / \sqrt{2}

⇒ Length of adjacent side/Hypotenuse = 1/√2

Therefore, we can write that cos (45°) = 1/√2

• Practical Method

You can also find the value of cos of angle 45° practically by constructing a right-angled triangle with 45° angle by geometrical tools.

Draw a straight horizontal line from Point I and then construct an angle of 45° using the protractor.

Set compass to any length by a ruler. Here, the compass is set to 6.5 cm. Now, draw an arc on the 45° angle line from point I and it intersects the line at point J.

Finally, draw a perpendicular line on the horizontal line from point J and it intersects the horizontal line at point K perpendicularly. Thus, a right-angled triangle ∆JIK is formed.

Now,  calculate the value of the cosine of 45 and for this, measure the length of the adjacent side by a ruler. You will observe that the length of the adjacent side is 4.6 cm. The length of the hypotenuse is taken as 6.5 cm in this example.

Now, find the ratio of lengths of the adjacent side to the hypotenuse and get the value of the cosine of angle 45°.

cos (45°) = IK/IJ = 4.6/6.5

So, cos (45°) = 0.7076923077… ≈ 0.7071

• Trigonometric Method

We can prove the value of cos (45°) with a trigonometric approach.

we know that, sin 45° = 1/√2

Also, by trigonometric identities,

\sin ^{2} x+\cos ^{2} x=1
Or \cos ^{2} x=1-\sin ^{2} x
Put x=45^{\circ}
\cos ^{2} 45^{\circ}=1-\sin ^{2} 45^{\circ}

Put the value of sin 45° 

\cos ^{2} 45^{\circ}=1-(1 / \sqrt{2})^{2}
\cos ^{2} 45^{\circ}=1-1 / 2
\cos \left(45^{\circ}\right)=\sqrt{1 / 2}=1 / \sqrt{2}

Hence, we proved the value of cos (45°) using different approaches.

Example

1. Evaluate: cos 45° + sin 45°

We know that cos (45°) = sin (45°) = 1/√2.
So, cos (45°) + sin (45°)
= 1/√2 + 1/√2
= 2/√2
= √2

2. Evaluate: 2 sin 45° – 4 cos 45°

We know that cos (45°) = sin (45°) = 1/√2.
So, 2 sin (45°) – 4 cos (45°)
= 2 (1/√2) – 4(1/√2)
= 2/√2 – 4/√2
= -2/√2
= -√2